Random homomorphisms into the orthogonality graph

TL;DR

This paper introduces a new approach to defining subgraph densities in orthogonality graphs on unit spheres, aiming to advance the understanding of limit objects in graph theory and related fields.

Contribution

It proposes a novel framework for subgraph densities in orthogonality graphs, addressing challenges in the 'middle' range and linking to applications in various scientific domains.

Findings

Defined subgraph densities in orthogonality graphs on spheres

Identified difficulties in the 'middle' range of subgraph densities

Suggested relevance to Markov spaces and applications in physics and information theory

Abstract

Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the "middle ranges", the notion of subgraph densities in these limit objects remains elusive. We define subgraph densities in the orthogonality graphs on the unit spheres in dimension $d$, under appropriate sparsity condition on the subgraphs. These orthogonality graphs exhibit the main difficulties of defining subgraphs the "middle" range, and so we expect their study to serve as a key example to defining subgraph densities in more general Markov spaces. The problem can also be formulated as defining and computing random orthogonal representations of graphs. Orthogonal representations have played a role in information theory, optimization, rigidity theory and quantum physics, so to study random ones may be of interest from the point of view of these applications as well.